1. **State the problem:** We need to find the measure of the minor arc JK in a circle where chords JK and NM intersect at the center L. 2. **Given information:** - Angle at center L is 92°. - Arc JM measures $(x + 15)^\circ$. - Arc NK measures $(2x - 5)^\circ$. 3. **Key formula:** The angle formed by two intersecting chords at the center of a circle equals half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Mathematically, for angle $\angle JLK$, $$\angle JLK = \frac{1}{2} (mJM + mNK)$$ 4. **Apply the formula:** $$92 = \frac{1}{2} ((x + 15) + (2x - 5))$$ 5. **Simplify inside the parentheses:** $$(x + 15) + (2x - 5) = 3x + 10$$ 6. **Rewrite the equation:** $$92 = \frac{1}{2} (3x + 10)$$ 7. **Multiply both sides by 2 to eliminate the fraction:** $$2 \times 92 = 3x + 10$$ $$184 = 3x + 10$$ 8. **Subtract 10 from both sides:** $$184 - 10 = 3x$$ $$174 = 3x$$ 9. **Divide both sides by 3:** $$x = \frac{174}{3}$$ $$x = 58$$ 10. **Find the measure of minor arc JK:** Since JK is the minor arc intercepted by chords J and K, and the arcs JM and NK are parts of the circle, the minor arc JK is the difference between arc JM and arc NK. Calculate arc JM: $$x + 15 = 58 + 15 = 73^\circ$$ Calculate arc NK: $$2x - 5 = 2(58) - 5 = 116 - 5 = 111^\circ$$ Since the circle's total arc is 360°, the minor arc JK is: $$360 - (73 + 111) = 360 - 184 = 176^\circ$$ **Final answer:** $$\boxed{176^\circ}$$